#6459997

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6,8,9)(7,10,11,12)(13,14,17,22,18,23,27,21)(15,24,25,28,26,19,16,20);(6,9)(10,12)(13,14)(15,20)(16,24)(17,21)(18,23)(19,25)(22,27)(26,28);(1,2,3,4)(13,15,17,25,18,26,27,16)(14,19,22,20,23,24,21,28);(5,7,8,11)(6,10,9,12)(13,16,17,15,18,25,27,26)(14,20,22,24,23,28,21,19);(13,17,18,27)(14,21,23,22)(15,25,26,16)(19,28,24,20)

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367193);